Chapter 6. Calculation
In the present chapter we are going to look at processes that the modern calcula-
tor has rendered obsolescent, that is, the basic operations of arithmetic: addition,
subtraction, multiplication, division, and the extraction of (square) roots. The
word obsolescent is used instead of the more emphatic obsolete because these pro-
cesses are still being taught to children in schools. But the skill that children are
acquiring becomes weaker with every passing year. In fact, it has been at least
30 years since high-school students were actually taught to extract a square root.
Even then, it was easier to consult a table of square roots than to carry out the
error-prone, complicated operation of finding the root. Of course, what has caused
these skills to fall out of use is the ready availability of hand-held calculators. This
latest technological marvel is a direct continuation of earlier technology to ease
the burden of concentration required in doing arithmetic, starting with counting
rods and counting boards, then moving on through the abacus and the slide rule.
The need to calculate has been a motivating force behind the development of me-
chanical methods of computation, and thus an important part of the history of
mathematics. In this chapter we look first at the earliest methods developed for
calculating, concentrating on multiplication and division (or, in the case of Egypt,
processes equivalent to these) and the extraction of roots. We shall also look at
three important motives for calculating: (1) commercial transactions involving la-
bor, construction, and trade; (2) geometric problems of area and volume involving
surveying and engineering; and (3) regulation of the calendar, especially finding
important dates such as Easter.
1. Egypt
The richest source of information on Egyptian methods of calculation is the Ahmose
(Rhind) Papyrus described in Chapter 2. After the descriptive title, the papyrus
begins with the table of numbers shown in Fig. 1 below. In the modern world, we
think of arithmetic as consisting of the four operations of addition, subtraction,
multiplication, and division performed on whole numbers and fractions. We learn
the rules for carrying out these operations in childhood and do them automatically,
without attempting to prove that they are correct. The situation was different for
the Egyptian. To the Egyptian, it seems, the fundamental operations were addition
and doubling, and these operations were performed on whole numbers and parts.
We need to discuss both the operations and the objects on which they were carried
out.
Let us consider first the absence of multiplication and division as we know
them. The tables you looked at in Problem 5.2 should have convinced you that
there is something special about the number 2. We don't normally say "one-twoth"129